1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding an orthonormal system

  1. Feb 5, 2006 #1
    Hey,
    I've taken linear algebra a long ago, and now came across a simple-looking question that i need help with

    Construct an orthonormal system from the three functions:
    1, x, 3x^2 - 1

    Can anyone give me a pointer to solving this question?
    Thanks a bunch!:blushing:
     
  2. jcsd
  3. Feb 5, 2006 #2
    Apply Gram-Schmidt?
     
  4. Feb 5, 2006 #3
    oh ya, this question is from a calculus homework before we get into Fourier Series and integral, so I don't think it wants us to apply Gram-Schmidt method.
     
  5. Feb 5, 2006 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    ortho, and normal, only make sense with respect to an inner product. You haven't given one. And just because you've not been taught the formal proof of Gram-Schmidt doesn't mean you're not supposed to figure it out on your own; it is merely projection formally written down and not at all a tricky thing to figure out on your own.
     
  6. Feb 5, 2006 #5
    it was with respect to 1...
     
  7. Feb 5, 2006 #6

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    1 what?
    I can think of many different inner products on function spaces, and polynomial rings, none of them is called 1.
     
  8. Feb 5, 2006 #7
    Sorry...Here's the entire question

    Show that 1, x, 3x^2 - 1 are orthogonal functions with respect to the weight function 1 on the interval [-1, 1]. Construct an orthonormal system from the three functions.


    I got the first part of the question, I'm stuck on the orthnormal part.

    Currently looking through linear algebra notes, not quite understanding how to do it using Gram-Schmidt.
     
  9. Feb 5, 2006 #8

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    You know how to do projection? Ie given vectors v and w write w as the sum of a vector parallel to v and one orthogonal to v. That's all gram schmidt is, but you just do it again, and again, and again.....
     
  10. Feb 5, 2006 #9
    I thought Gram-Schmidt is used to find orthogonal basis only.
    Sorry I might sound dumb, but the last time I took linear algebra was 2 years ago...don't remember much from it.
     
  11. Feb 5, 2006 #10

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    You can't get from a set of orthogonal vectors to a set of orthogonal vectors all of length one?
     
  12. Feb 5, 2006 #11
    how do we do that?
     
  13. Feb 5, 2006 #12

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    If I were to say here are u and v two orthogonal vectors (none of which is zero), you can now write down two orthogonal vectors of length 1, right? I can't tell if you're kidding me because you think I"m being patronizing or if you can't find a vector of length one from another vector (it has been some years you say since you did this).
     
  14. Feb 5, 2006 #13
    :bugeye: I'm not a very wordy person and I learn from seeing equations and numbers and examples, I guess it's kinda hard to explain it like that. Thanks for your help, I'll think about it for now.

    according to my notes, it says that a system of orthogonal functions w.r.t. weight q of [a, b] is also orthonormal if (see attachment) for all m. Is that what you meant?
     

    Attached Files:

    Last edited: Feb 5, 2006
  15. Feb 5, 2006 #14

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    If v is a vector (not the zero vector) then v/|v| is a unit vector in the same direction.
     
  16. Feb 5, 2006 #15
    So if i were to rewrite the 3 functions in terms of vectors, would they become like this?

    0 0 1 <-- 1
    0 1 0 <-- x
    3 0 -1 <-- 3x^2-1

    so 1st vector would be
    0
    0
    3
    ?
     
  17. Feb 5, 2006 #16
    Ok, I finally understand the part after the unit vector.
    What is the next step once I've got the unit vector for each of the function?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Finding an orthonormal system
  1. Orthonormal system (Replies: 1)

Loading...